Download counting, calculating, representing

COUNTING, CALCULATING, REPRESENTING
WRITING AND READING SEXAGESIMAL NUMBERS IN ANCIENT MESOPOTAMIA
Grégory Chambon
INTRODUCTION
The
expression
“sexagesimal
numbers”
brings
to
mind
to
the
modern division of the circle and the hour, as well as to a long
history beginning with the Babylonians and going on with the Greeks
and the Arabs. Numerous works concerning the history of mathematic
have repeated that the “Babylonians counted in a sexagesimal system
with place value notation”, in the same way as our modern decimal
system,
and
numeration,
that
used
this
by
system
the
has
been
astronomers
an
until
academic
the
system
period
of
of
the
Renaissance.
The cuneiform documentation offers us a terminus ante quem for
the use of the sexagesimal numbers in place value notation. Until now,
a “scratch pad” from the Ur III period (end of the third Millennium)
represents the earliest known and securely dated example of this use.
Nevertheless,
system
is
the
origin
still
much
of
the
so-called
debated
both
sexagesimal
by
place
philologists
value
and
by
mathematicians. Controversy over its origin revolves usually around
three points:
1- Several authors have focused their attention on the mutually
dynamic relationship between writing and language. How and in which
language were the numbers notations pronounced? By recognizing the
linguistic nature of the problem, it would be possible to explain the
origin
of
counting
with
sixty.
As
sexagesimal
notations
can
be
discerned in the earliest written documents from Uruk at the end of
the fourth millennium, it is debated whether the Sumerian numbers
words originated in these notation or whether it is the other way
round. The study of the linguistic background brings us therefore to
the so-called “Sumerian question”: Was the language of the earliest
texts from Mesopotamia actually Sumerian?
2- A graphic explanation for the invention of the sexagesimal
place value notation would be that the sign for sixty came to be
written with the standard stylus in the same manner as the sign for
one, that is to say, with a cuneiform wedge. This lack of visual
distinction
during
a
long
period
has
been
interpreted
as
the
fundamental step of the development of the principle of sexagesimal
place
value
notation.
This
assumption
– 1 –
requires
a
mathematical
or
anthropological explanation to the sexagesimal pattern of the system.
Why has a system of numerical notation, which rises in powers of
sixty, played a prominent role during two millennia?
3- Others scholars try to advance or to infirm the thesis that
sexagesimal numbers in place value notation must have been used in the
solution
procedures
-
which
were
never
explicit
–
underlying
mathematical problems from the 3rd millennium. In which respect would
the formal structure of the problems themselves, with the measures for
lengths,
areas,
capacities
etc.,
reflect
the
recourse
to
the
sexagesimal place value system?
On one hand, these various methodologies success in showing that
the problem of the origin cannot be solved within the limited point of
you of a single field. On the other hand, some assumptions they are
based on raises a number of fundamental problems, which I propose to
discuss in my paper. My purpose is not to give easy answers to these
questions and to take part on all the debates, but rather to address
new questions concerning the study of the origin of the sexagesimal
place value system, which will take into account the functions of
writing as a cultural technique: In which context and how were the
different sexagesimal systems of numbers notations actually used? What
does
"reading"
a
written
number
mean?
How
heuristic role by representing numbers on clay?
– 2 –
does
writing
play
a